American Mathematical Monthly -November 2007

November 2007

Random Walker Ranking for NCAA Division I-A Football
By: Thomas Callaghan, Peter J. Mucha, and Mason A. Portertscallag@stanford.edu, mucha@unc.edu, mason@its.caltech.edu
Each December, college football fans and pundits across America debate which two teams should meet in the NCAA Division I-A National Championship game. The Bowl Championship Series (BCS) standings employed to select the teams invited to this game are intended to provide an unequivocal #1 v. #2 game for the championship; however, this selection process has itself been highly controversial in recent years. The computer algorithms that constitute one part of the BCS standings often act as lightning rods for the controversy, in part because they are inadequately explained to the public. We present an alternative algorithm that is simply explained yet remains effective at ranking the best teams. We define a ranking in terms of biased random walkers on the graph formed by the schedule of games played, with two teams (vertices) connected by an edge if they played each other. Each random walker moves from team to team by selecting a game and "voting" for its winner with probability $p$, tracing out a never-ending path motivated by the "my team beat your team" argument. We study the statistical properties of a collection of such walkers, relate the rankings to the community structure of the underlying network, and demonstrate the results for recent NCAA Division I-A seasons. We also discuss the algorithm's asymptotic behavior, illustrated with some analytically tractable cases for round-robin tournaments, and discuss possible generalizations.

Solving Differential Equations by Symmetry Groups
By: John Starrettjstarret@nmt.edu
Although the best-known mathematical applications of Sophus Lie's theory of continuous groups are in differential geometry and control theory, there is now a renewed interest in his original application to solutions of differential equations. Recently a number of fine texts have appeared, written at various levels, and the Maple computer algebra system incorporates a nice package for using Lie's methods to solve differential equations. This article is a simple introduction to the method applied to first-order ordinary differential equations, emphasizing the geometric aspects of the theory.

The Map-Coloring Game
By: Tomasz Bartnicki, Jaroslaw Grytczuk, H. A. Kierstead, and Xuding Zhut.bartnicki@wmie.uz.zgora.pl, j.grytczuk@wmie.uz.zgora.pl, kierstead@asu.edu, zhu@math.nsysu.edu.tw
Suppose that Alice wants to color a planar map using four colors in a proper way, that is, so that any two adjacent regions get different colors. She asks Bob to help her in this task. They color the regions of a map alternately, with Alice going first. Bob agrees to cooperate by respecting the rule of a proper coloring. However, for some reason he does not want the job to be completed - his secret aim is to achieve a "bad" partial coloring, one that cannot be extended to a proper coloring of the whole map. Is it possible for Alice to complete the coloring somehow, in spite of Bob's insidious plan? If not, then how many additional colors are needed to guarantee that the map can be successfully colored, no matter how clever Bob is?